Characterization of weak-star–weak points of continuity for nonseparable duals

Characterize the weak-star–weak points of continuity of the dual unit ball of the Bochner space Lp(μ, X) in terms of the weak-star–weak points of continuity of the dual unit ball of X for 1 < p < ∞ when the Banach space X has a nonseparable dual.

Background

The paper proves, under separability assumptions on X or X∗, characterizations relating weak-star–weak points of continuity of the dual unit ball of Lp(μ, X) to the corresponding pointwise property in X∗. In particular, for 1 < p < ∞, the relevant pointwise condition is that h(t)/∥h(t)∥ be a weak-star–weak point of continuity of S(X∗) almost everywhere where h(t) is nonzero.

Question 3.6 asks whether an analogous characterization remains valid when X∗ is not separable. This is unresolved because the paper’s principal arguments rely on separability, including measurable-selection and measurability tools used to pass between pointwise and Bochner-space properties.